PHY5210 W15

Reviewed solutions to problem set #9.

Chapter 15: Special Relativity

Reading

Taylor 15.1 --15.4 for today and Taylor 15.5 -- 15.7 for Wednesday.

Recall

Introduction to Special Relativity

Special relativity can be treated on a number of levels. I will try to present special relativity in a way that is consistent with the level of the course to date. This will also be at the level for an introduction to general relativity. General relativity is beyond the scope of most undergraduate curricula, but many people are interested to learn the basics, and I hope this approach to special relativity makes that easier.

The difference between special and general relativity is that general relativity deals with gravity while special relativity doesn't. This seems somewhat trivial, but incorporating gravity brings a whole mathematical framework and numerous interesting phenomena.

The Laws of Special Relativity

The implications of special relativity can be derived from a couple of laws, and clever thinking. First, a definition of an inertial frame that is more appropriate for this work.

Definition of an Inertial Frame

An inertial frame is any reference frame (that is, a system of coordinates x, y, z and time t) in which all the laws of physics hold in their usual form.

This definition is more encompassing than Newton's first law, as it applies to the laws of electromagnetism and quantum mechanics as well as mechanics. Of course, these other laws of physics were unknown in Newton's time.

First Law of Relativity

If S is an inertial frame and if a second frame S' moves with constant velocity relative to S, then S' is also an inertial frame.

Second Law of Relativity

The speed of light (in vacuum) has the same value c in every direction in all inertial frames.

This law has been tested to considerable precision and has never been falsified.

There are alternative ways to specify the laws of relativity that lead to the same results. For instance, David Mermin has proposed an alternative form of the second law.

Second Law of Relativity (Alternate)

There is a maximum speed vmax of motion of any object in any frame.

The maximum speed is equal to the speed of light in vacuum, vmax = c. We will follow Taylor and use the first form, but it is interesting to know that there are equally valid starting points.

The Lorentz Transformation

In E&M we find the wave equation for light to be the pair of partial differential equations

∇²E = (1/c²)(∂²E/∂t²)    and    ∇²B = (1/c²)(∂²B/∂t²)

or, limiting ourselves to light propagating along the z direction

∂²Ex/∂z² = (1/c²)(∂²Ex/∂t²)    and    ∂²Bx/∂z² = (1/c²)(∂²Bx/∂t²),

and likewise for the y components, where c is the speed of light (≈ 3×108m/s). Now imagine the light viewed from a different reference frame moving with respect to the first along the z direction with speed v. We relate the coordinates in the new frame (primed) to those in the original (unprimed) by the Galilean transformation, z' = z - vt and t' = t (so z = z' + vt'). Let ψ denote any of the fields Ex, Ey, Bx, or By. To see how the wave equation transforms, we can go through the derivatives piece by piece, starting with the z derivative

∂ψ/∂z = (∂ψ/∂z')(∂z'/∂z + ∂z'/∂t) = (∂ψ/∂z')(1 - v)

Differentiating a second time yields the desired second derivative

∂²ψ/∂z² = (∂ψ/∂z')(∂z'/∂z + ∂z'/∂t) = (∂²ψ/∂z'²)(1 - v)²

Follow the same route for the time derivative

∂ψ/∂t = (∂ψ/∂t')(∂t'/∂t) = ∂ψ/∂t'

Differentiating a second time yields the desired second derivative

∂²ψ/∂t² = ∂²ψ/∂t'²

Putting these together, we see that the wave equation transforms as

(∂²ψ/∂z²)(1-v)² = (1/c²)(∂²ψ/∂t²),

for any of the electric or magnetic field components. The extra piece that is carried along doesn't cancel out, and presents a problem. While Newton's laws appear the same in inertial reference frames differing by a Galilean transformation, Maxwell's equations do not. Either Maxwell's equations are not the most general form for the equations of electromagnetism, or the Galilean transformation is not the correct way to connect inertial frames. There is strong evidence that Maxwell's equations are correct, so let's look at modifying the Galilean transformation.

The transformation that leaves Maxwell's equations invariant is called the Lorentz transformation. For the reference frames used above, it has the form z' = (z - vt)/√(1 - v²/c²) = γ(z - βct) and t' = (t - vz/c²)/√(1 - v²/c²) = γ(t - βz/c) where β = v/c and γ = (1 - β²)-½. Now the terms of the wave equation transform as

∂ψ/∂z = (∂ψ/∂z')(∂z'/∂z) + (∂ψ/∂t')(∂t'/∂z) = (∂ψ/∂z')γ - (∂ψ/∂t')γβ/c
∂²ψ/∂z² = (∂²ψ/∂z'²)γ² + (∂²ψ/∂t'²)γ²β²/c² - 2(∂²ψ/∂z'∂t')γ²β/c

and

∂ψ/∂t = (∂ψ/∂z')(∂z'/∂t) + (∂ψ/∂t')(∂t'/∂t) = -(∂ψ/∂z')γβc + (∂ψ/∂t')γ
∂²ψ/∂t² = (∂²ψ/∂z'²)γ²β²c² + (∂²ψ/∂t'²)γ² - 2(∂²ψ/∂z'∂t')γ²βc.

Under the Lorentz transformation, the wave equation transforms as

(∂²ψ/∂z²) - (1/c²)(∂²ψ/∂t²) = (∂²ψ/∂z'²)γ²(1 - β²) + (∂²ψ/∂t'²)γ²/c²(β² - 1) - 2(∂²ψ/∂z'∂t')γ²β/c (1 - 1)

This simplifies to

(∂²ψ/∂z²) - (1/c²)(∂²ψ/∂t²) = (∂²ψ/∂z'²) - (1/c²)(∂²ψ/∂t'²).

Although I won't show it, the Lorentz transformation leaves Maxwell's equations (in vacuum) unchanged in form, not just the wave equation derived from them.

The laws of mechanics are well tested for motion on earth, and orbits of planets around the sun, and the predictions derived from them are in good agreement with measurements. Maxwell's equations are also well tested, and, if anything, the predictions are in even better agreement with measurements. That is, we are able to test the laws of electromagnetism to greater levels of precision than the laws of mechanics. At the velocities normally encountered in mechanics, the Lorentz transformation is little different from the Galilean transformation -- the Galilean transformation and non-relativistic mechanics can be thought of as the lowest order term in an expansion of the full relativistic form.

© 2015 Robert Harr